Problem 19 - Calculus and its applications

Question

Note that \(x\) represents the actual year in following table. $$ \begin{array}{|l|c|} \hline \multicolumn{1}{|c|} {\text { YEAR, } \boldsymbol{x}} & \begin{array}{c} \text { WORLD RECORD, } \boldsymbol{y} \\ \text { (in minutes:seconds) } \end{array} \\ \hline 1875 \text { (Walter Slade) } & 4: 24.5 \\ 1894 \text { (Fred Bacon) } & 4: 18.2 \\ 1923 \text { (Paavo Nurmi) } & 4: 10.4 \\ 1937 \text { (Sidney Wooderson) } & 4: 06.4 \\ 1942 \text { (Gunder Hagg) } & 4: 06.2 \\ 1945 \text { (Gunder Hagg) } & 4: 01.4 \\ 1954 \text { (Roger Bannister) } & 3: 59.6 \\ 1964 \text { (Peter Snell) } & 3: 54.1 \\ 1967 \text { (Jim Ryun) } & 3: 51.1 \\ 1975 \text { (John Walker) } & 3: 49.4 \\ 1979 \text { (Sebastian Coe) } & 3: 49.0 \\ 1980 \text { (Steve Ovett) } & 3: 48.40 \\ 1985 \text { (Steve Cram) } & 3: 46.31 \\ 1993 \text { (Noureddine Morceli) } & 3: 44.39 \end{array} $$ a) Find the regression line, \(y=m x+b,\) that fits the data in the table. (Hint: Convert each time to decimal notation; for instance, \(\left.4: 24.5=4 \frac{24.5}{601}=4.4083 .\right)\) b) Use the regression line to predict the world record in the mile in 2015 and 2020 . c) In July 1999 , Hicham El Guerrouj set the current (as of July 2014 ) world record of 3: 43.13 for the mile. (Sources: USA Track \& Field and infoplease.com.) How does this compare with what is predicted by the regression line?

Step-by-Step Solution

Answer

The regression line is formulated, and predictions for years 2015 and 2020 are made. The predicted times are compared with the 1999 world record set by Hicham El Guerrouj.

Step 1: Convert time to decimal format

First, we need to convert each world record time from minutes:seconds format into decimal minutes. For example, 4:24.5 becomes \(4 + \frac{24.5}{60} = 4.4083\). Apply the same method to all the other times to populate the table with decimal values.

Step 2: Calculate sums for regression line

For linear regression, we need sums of the years \(x\), world record times \(y\), \(xy\), and \(x^2\). Calculate these sums using the converted decimal values.

Step 3: Determine the slope (m)

Use the formula \( m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \), where \(n\) is the number of data points. Substitute the sums calculated in Step 2 to find the slope \(m\).

Step 4: Determine the y-intercept (b)

Calculate \(b\) using the formula \( b = \frac{(\sum y) - m(\sum x)}{n} \). Substitute \(m\) from Step 3 and the other sums to find the y-intercept \(b\).

Step 5: Formulate the regression equation

Combine the slope \(m\) and y-intercept \(b\) into the regression line equation \(y = mx + b\).

Step 6: Predict world records for 2015 and 2020

Substitute \(x = 2015\) and \(x = 2020\) into the regression line equation to predict the world record times for these years.

Step 7: Compare predicted and actual records for 1999

Using the regression line equation, calculate the predicted world record time for 1999. Compare this prediction with the actual record of 3:43.13 set by Hicham El Guerrouj.

Key Concepts

Data Conversion

In this exercise, an essential first step is data conversion. This involves changing the format of each world record time from minutes and seconds into simple decimal numbers. For instance, take 4:24.5, a time elapsed in minutes and seconds. First, convert the seconds into decimal form by dividing them by 60. So, 24.5 seconds becomes 24.5/60. Add this result to the minutes to get a decimal time: 4 + 24.5/60 = 4.4083. This process must be completed for each record listed in the table.

Performing this conversion is crucial for accurate calculations in linear regression, which requires numerical data in a linear format. By standardizing units, you ensure consistency, allowing for clearer analysis during future steps. With the data now in a suitable form, you can proceed to the next steps with confidence.

World Record Prediction

Predicting future world records involves a detailed process using the regression equation. The regression line, formulated from past records, indicates a trend over time. To forecast records for future years, substitute the relevant year values into the regression equation. In this exercise, the years 2015 and 2020 are of particular interest.

By inserting these year values into the equation, we obtain hypothetical record times under the assumption that the established trend continues. This can be an exciting exercise, as it provides a quantitative glimpse into the future, though it is important to remember that real-world outcomes can deviate from predictions due to unforeseen factors.

Regression Equation

The regression equation is a central player in conducting a linear regression analysis. It takes the form of a line: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To understand this more thoroughly, the slope \(m\) represents the rate at which world records are improving each year.

Determining the slope involves several key calculations, including the sums of years, times, products of these values, and squares of the years. Once calculated, the slope reveals how quickly or slowly records change over time.

The y-intercept \(b\) is the point where the line crosses the y-axis. This value is determined by using the known variables and calculated slope in the formula: \( b = \frac{(\sum y) - m(\sum x)}{n} \). With both \(m\) and \(b\) in hand, the regression equation offers a powerful tool for predicting future trends.

Regression Analysis

Regression analysis examines relationships between variables. In this context, it focuses on the relationship between the years and the world record times. The main goal is to find a line that best predicts future world records based on past data trends.

Conducting regression analysis requires calculating certain sums and values from the dataset. By finding the optimal slope and intercept, you establish the basis for the regression line. This line offers a model for how one variable affects the other, providing a predictive snapshot.

This type of analysis is useful for making informed projections not just in athletics but across many fields. Understanding how to perform regression analysis opens up opportunities for predicting and planning, by harnessing the power of historical data.